Home > Functional Analysis > Conditions for an operator to be bounded

Conditions for an operator to be bounded


Let E be a Banach space, and let T : E \to E^* be a linear operator. In each of the following cases, prove that T is bounded.
a) \langle Tx,x \rangle \geq 0,\ \forall x \in E;
b) \langle Tx,y \rangle=\langle Ty,x\rangle,\ \forall x,y \in E.

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  1. Diana
    February 21, 2011 at 10:03 pm

    Spatiul nu e cumva Hilbert?

    • February 21, 2011 at 10:19 pm

      Nu. E corect asa cum e. Problema e din cartea lui Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer 2010.
      Semnificatia e urmatoarea, pentru o functionala f \in E^* se noteaza f(x)=\langle f, x \rangle.

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